Standard monomials and extremal point sets

Abstract

We say that a set system F⊆2[n] shatters a set S⊆[n] if every possible subset of S appears as the intersection of S with some element of F, and we denote by Sh(F) the family of sets shattered by F. According to the Sauer–Shelah lemma we know that in general, every set system F shatters at least |F| sets, and we call a set system shattering-extremal if |Sh(F)|=|F|. In Mészáros (2010) and Rónyai and Mészáros (2011), among other things, an algebraic characterization of shattering-extremality was given, which offered the possibility to generalize the notion to general finite point sets. Here we extend the results obtained for set systems to this more general setting, and as an application, strengthen a result of Li, Zhang and Dong from Li et al. (2012).